On a Mathematical Method for Discovering Relations Between Physical Quantities: Maxwell`s equations revisited
Philippe Chévalier · Ghent University Academic Bibliography (Ghent University) · 2014
Quantity calculus defines the rules that apply to SI physical quantities used in physics and engineering.This research aims at the development of a mathematical method for discovering the mathematical form of the relations between physical quantities.Laws of physics are unique relations obeying unknown mathematical selection rules.Here, we show that each SI physical quantity, that is represented by a lattice point in a seven dimensional integer lattice, has a unique 7Dhypersphere.The lattice points incident on the 7D-hypersphere are forming rectangles containing the origin o, the lattice point z representing the selected physical quantity and the lattice point representations x, y of a pair of distinguishable physical quantities [x], [y] where z = x + y .The resulting rectangles are the geometric representations of the realizable binary form equations for the selected physical quantity [z] .The isoperimeter distribution of the resulting rectangles shows the exceptional occurrence of unique rectangles that can be associated with unique relations between physical quantities.We find unknown integer sequences representing the number of unique rectangles and the number of nondegenerated rectangles formed by 4 lattice points o, x, y, z in Z 7 as function of the infinity norm ∥z∥∞ = s .The ratio of the number of unique rectangles to the number of rectangles is decreasing for increasing infinity norm ∥z∥∞ = s .We apply the ``hypersphere method´´on the physical quantities E, H, D, B to validate the mathematical method and find the integral forms of Maxwell's equations.A second method is developed for n-ary form equations based on the Gödel encoding of a leader class of a physical quantity.The canonical factorization of the Gödel number in n distinct factors generates the n-ary form equations of a physical quantity.